Symmetric and asymmetric solitons in dual-core couplers with competing quadratic and cubic nonlinearities

Abstract

We consider the model of a dual-core spatial-domain coupler with χ(2) and χ(3) nonlinearities acting in two parallel cores. We construct families of symmetric and asymmetric solitons in the system with self-defocusing χ(3) terms and test their stability. The transition from symmetric to asymmetric soliton branches and back to the symmetric ones proceeds via a bifurcation loop. Namely, a pair of stable asymmetric branches emerges from the symmetric family via a supercritical bifurcation; eventually, the asymmetric branches merge back into the symmetric one through a reverse bifurcation. The existence of the loop is explained by means of an extended version of the cascading approximation for the χ(2) interaction, which takes into regard the cross-phase modulation part of the χ(3) interaction. When the intercore coupling is weak, the bifurcation loop features a concave shape, with the asymmetric branches losing their stability at the turning points. In addition to the two-color solitons, which are built of the fundamental-frequency (FF) and second-harmonic (SH) components, in the case of the self-focusing χ(3) nonlinearity we also consider single-color solitons, which contain only the SH component but may be subject to the instability against FF perturbations. Asymmetric single-color solitons are always unstable, whereas the symmetric ones are stable, provided that they do not coexist with two-color counterparts. Collisions between tilted solitons are studied, too.